When Less Is More: The Case for Teaching Less Math in School

In 1929, the superintendent of schools in Ithaca, New York, sent out a challenge to his colleagues in other cities. "What," he asked, "can we drop from the elementary school curriculum?" He complained that over the years new subjects were continuously being added and nothing was being subtracted, with the result that the school day was packed with too many subjects and there was little time to reflect seriously on anything. This was back in the days when people believed that children shouldn't have to spend all of their time at school work--that they needed some time to play, to do chores at home, and to be with their families--so there was reason back then to believe that whenever something new is added to the curriculum something else should be dropped.

One of the recipients of this challenge was L. P. Benezet, superintendent of schools in Manchester, New Hampshire, who responded with this outrageous proposal: We should drop arithmetic! Benezet went on to argue that the time spent on arithmetic in the early grades was wasted effort, or worse. In fact, he wrote: "For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning facilities." All that drill, he claimed, had divorced the whole realm of numbers and arithmetic, in the children's minds, from common sense, with the result that they could do the calculations as taught to them, but didn't understand what they were doing and couldn't apply the calculations to real life problems. He believed that if arithmetic were not taught until later on--preferably not until seventh grade--the kids would learn it with far less effort and greater understanding.[1]

Think of it. Today whenever we hear that children aren't learning much of what is taught in school the hue and cry from the educational establishment is that we must therefore teach more of it! If two hundred hours of instruction on subject X does no good, well, let's try four hundred hours. If children aren't learning what is taught to them in first grade, then let's start teaching it in kindergarten. And if they aren't learning it in kindergarten, that could only mean that we need to start them in pre-kindergarten! But Benezet had the opposite opinion. If kids aren't learning much math in the early grades despite considerable time and effort devoted to it, then why waste time and effort on it?

Benezet followed his outrageous suggestion with an outrageous experiment. He asked the principals and teachers in some of the schools located in the poorest parts of Manchester to drop the third R from the early grades. They would not teach arithmetic--no adding, subtracting, multiplying or dividing. He chose schools in the poorest neighborhoods because he knew that if he tried this in the wealthier neighborhoods, where parents were high school or college graduates, the parents would rebel. As a compromise, to appease the principals who were not willing to go as far as he wished, Benezet decided on a plan in which arithmetic would be introduced in sixth grade.

As part of the plan, he asked the teachers of the earlier grades to devote some of the time that they would normally spend on arithmetic to the new third R--recitation. By "recitation" he meant, "speaking the English language." He did "not mean giving back, verbatim, the words of the teacher or the textbook." The children would be asked to talk about topics that interested them--experiences they had had, movies they had seen, or anything that would lead to genuine, lively communication and discussion. This, he thought, would improve their abilities to reason and communicate logically. He also asked the teachers to give their pupils some practice in measuring and counting things, to assure that they would have some practical experience with numbers.

In order to evaluate the experiment, Benezet arranged for a graduate student from Boston University to come up and test the Manchester children at various times in the sixth grade. The results were remarkable. At the beginning of their sixth grade year, the children in the experimental classes, who had not been taught any arithmetic, performed much better than those in the traditional classes on story problems that could be solved by common sense and a general understanding of numbers and measurement. Of course, at the beginning of sixth grade, those in the experimental classes performed worse on the standard school arithmetic tests, where the problems were set up in the usual school manner and could be solved simply by applying the rote-learned algorithms. But by the end of sixth grade those in the experimental classes had completely caught up on this and were still way ahead of the others on story problems.

In sum, Benezet showed that kids who received just one year of arithmetic, in sixth grade, performed at least as well on standard calculations and much better on story problems than kids who had received several years of arithmetic training. This was all the more remarkable because of the fact that those who received just one year of training were from the poorest neighborhoods--the neighborhoods that had previously produced the poorest test results. Why have almost no educators heard of this experiment? Why isn't Benezet now considered to be one of the geniuses of public education? I wonder. [Note: Benezet's work was brought to my attention in a comment that Tammy added to my Feb. 24 post. Thanks, Tammy.]

For decades since Benezet's time, educators have debated about the best ways to teach mathematics in schools. There was the new math, the new new math, and so on. Nothing has worked. There are lots of reasons for this, one of which is that the people who teach in elementary schools are not mathematicians. Most of them are math phobic, just like most people in the larger culture. They, after all, are themselves products of the school system, and one thing the school system does well is to generate a lasting fear and loathing of mathematics in most people who pass through it. No matter what textbooks or worksheets or lesson plans the higher-ups devise for them, the teachers teach math by rote, in the only way they can, and they just pray that no smart-alec student asks them a question such as "Why do we do it that way?" or "What good is this?" The students, of course, pick up on their teachers' fear, and they learn not to ask or even to think about such questions. They learn to be dumb. They learn, as Benezet would have put it, that a math-schooled mind is a chloroformed mind.

In an article published in 2005, Patricia Clark Kenschaft, a professor of mathematics at Montclair State University, described her experiences of going into elementary schools and talking with teachers about math. In one visit to a K-6 elementary school in New Jersey she discovered that not a single teacher, out of the fifty that she met with, knew how to find the area of a rectangle.[2] They taught multiplication, but none of them knew that multiplication is used to find the area of a rectangle. Their most common guess was that you add the length and the width to get the area. Their excuse for not knowing was that they did not need to teach about areas of rectangles; that came later in the curriculum. But the fact that they couldn't figure out that multiplication is used to find the area was evidence to Kenschaft that they didn't really know what multiplication is or what it is for. She also found that although the teachers knew and taught the algorithm for multiplying one two-digit number by another, none of them could explain why that algorithm works.

The school that Kenschaft visited happened to be in a very poor district, with mostly African American kids, so at first she figured that the worst teachers must have been assigned to that school, and she theorized that this was why African Americans do even more poorly than white Americans on math tests. But then she went into some schools in wealthy districts, with mostly white kids, and found that the mathematics knowledge of teachers there was equally pathetic. She concluded that nobody could be learning much math in school and, "It appears that the higher scores of the affluent districts are not due to superior teaching but to the supplementary informal ‘home schooling' of children." [Note: A reference to Kenschaft's article was provided to me by Sue VanHattum, who writes a great blog called "Math Mamma Writes."]

At the present time it seems clear that we are doing more damage than good by teaching math in elementary schools. Therefore, I'm with Benezet. We should stop teaching it. In my next post--about two weeks from now--I'm going to talk about how kids who don't go to traditional schools learn math with no or very little formal instruction. If you have a story to tell me about such learning, which might contribute to that post, please tell it in the comments section below or email it to me at grayp@bc.edu within the next week.[3] I've already collected quite a few such stories, but the more I receive the more I'll have to say.

[3] If you email me, please forgive me if I take a while to respond to your email. My box has been very full lately. If I use your story, I will use your first name only and will not use the name of the child you write about.

This problem is enormous. Teachers don't understand the material they are teaching, and so they can't answer questions from intellectually curious students. Instead, they shut down that kind of thought.

Case in point:

Some years ago, my (now ex-) wife was involved in a "trivia night" fundraiser at her elementary school, and they wanted me on their "teacher team" to round out their knowledge. They had almost everything covered except some technology-related topics and I was an IT guy. In round four, my moment to shine arrived, as the category was "Math & Science" and one of the questions was, "give the first five digits of pi." I quickly said, "3.1415." The 9 teachers at the table ignored me and wrote down "22/7" on scrap paper and began to divide it out. I observed this quietly at first, assuming that 22/7ths gave the right answer for the first 5 digits, but it doesn't. It gives something like 3.1427. I said, "Whoops, that won't work." They ignored me and consulted among themselves, concluding that they had all done the division properly on 22/7ths out to five digits. I said, "That's not right, it's 3.1415."

They shot me insolent looks and one said, "We're math teachers. This is what the book says and this is what we're teaching the students." I said, "The TEXTBOOK in school says that?" "Yes, it does." I thought, "No wonder we graduate generations of ignorant boobs, the book is wrong." But the more I thought about it, the more I found it unlikely that the book contained this error. They wrote down "3.1427" or whatever on our answer sheet and I doggedly insisted that their answer was wrong. I explained that pi was equal to the ratio of a circle's circumference to its diameter, and after an uncertain pause, one of the teachers summoned this fact from her vague recollection of her lesson plan: "Pi is an irrational number, so it can't be a ratio." The other teachers at the table exchanged knowing glances and murmured with approval with that "A-ha! Gotcha!" look on their faces, as they recalled that same fact and found their colleague's logic unimpeachable.

I pointed out that an irrational number was a number that could not be expressed as a ratio of integers, and that 22/7 was a ratio of integers, and so it was THEM who were clearly wrong, because the circumference and diameter of a circle are real numbers, not integers. There was apparently some confusion since they weren't entirely sure what an integer was, but when one of them remembered she triumphantly told me that she just had her class full of 5th graders calculate the area of a circle whose diameter was 10, and 10 is an integer, so HA! At this point I realized I was in the company of idiots. My ex-wife, who was extremely argumentative with me most of the time, wisely kept quiet. She knew I was smarter than her cohorts and probably right.

The debate continued, and one of the teachers finally cut me off and said, "I'm sorry, what's your background?" I said, "I'm a software engineer." The teacher said, "Degree?" I said, "English." She smiled and said, "I can settle this." She called over a friend who was collecting answer sheets, and said to the friend, "Sally, this gentleman thinks pi is not equal to 22/7, what do you think?" Sally looked at me with sympathy and said, "Of course it is. I just read that in Joseph's book the other day." (Joseph is her son at the school). I insisted they were wrong and explained why, and while she looked confused and insecure upon hearing my explanation, she finally cut me off and said, "I'm sorry, but I'm a civil engineer, and math is my job. Pi is 22/7ths."

At this point I began to lose my patience. I said, "There is no way that textbook says it's 22/7ths. Somebody go get a copy." My wife volunteered to run to her classroom and retrieve the textbook.

Despite Sally's authority, I basically bullied the teachers into begrudgingly writing down my answer, and we turned it in. When the round finished, the MC announced the answer, and the answer was that pi is equal to 22/7ths, or 3.1427 or whatever. They all turned to me with furious indignation and practically screamed, "WE TOLD YOU!!!!!"

I gritted my teeth and said, "They. Are. WRONG. Who wrote the questions?" They said, "Sally!"

God save us.

A great murmur arose at a few other tables as well, and my wife returned with the text book. The teacher with whom I had been butting heads the most grabbed it, thumbed through to the area of a circle chapter, jabbed her finger at a sentence, and shoved the book across the table at me. The sentence said, "Pi is an irrational number approximately equal to 22/7ths." The teacher sat back, crossed her arms triumphantly, and said, "I hope they taught you to read with your English degree." I read it, and slid the book back across and said, "They also taught us to read the footnotes." As she read looked back at the page, the blood drained from her face. At the end of the sentence was a little "1" in superscript, and dutifully noted in the footnote were the first 20 digits of the actual value of pi.

Of which the first five are 3.1415.

As if on cue, the MC then said, "We've got a correction on the 'pi' question, apparently there's been confusion, but we will now be accepting 3.1415 as a correct answer as well." I argued that 3.1427 was NOT CORRECT and shouldn't be accepted, but I was advised by my wife to accept the victory, such as it was, and let it go. Our table sat in sullen silence for a few moments, with the open math book still open before us.

I couldn't help myself. I said, "So, you teach math to future civil engineers, huh?"

It was a funny story I used to enjoyed telling, until several years later when a bridge spontaneously collapsed in Minneapolis, killing a half dozen people.

Your story about teachers absolutely convinced that pi was 22/7 reminded me of a chemistry teacher I used to work with at the Fieldston School in New York City some years ago. First, in case you don't know, Fieldston was and is an elite school among elite schools. This is the exact opposite of the "poor" schools that have been discussed here. The chemistry teacher that I worked with there calmly explained to me one afternoon (I was a physics teacher there at the time) that pi was exactly 22/7, and he used this as a bonus question on his chemistry exam! He knew this mathematical trivia with great certainty just like the teachers you encountered. This does not make him an idiot or them "idiots". They and he were seriously deluded by "common knowledge" acquired outside the textbooks. This pi problem appears to have been some sort of math "urban legend" for a while. The textbooks were all very clear that the value of 22/7 was only an approximation for pi, but somehow teachers and students were telling each other "off-textbook" that 22/7 really was pi. In other words, he didn't just think that this was a calculational convenience. He had learned the "legend" somewhere that pi, the quitessential irrational, transcendental number, was actually equal to a simple fraction. It took a bet of buying beers for the evening to convince him to look this up --and yes, he was quite sure, absolutely sure, that I would be buying beers in the end. He was genuinely, profoundly upset when the truth finally dawned on him. And I must re-iterate that this was an intelligent, well-educated science teacher working at a top-of-the-line private secondary school. Imagine how quickly this issue could be resolved today at the click of a mouse or the flick of a finger across a touchscreen.

Just thought I should add that to your tale of pi problems. It wasn't just them...

I see this is an old article, but I love this book by Jeanne Bendick, Archimedes and the Door of Science, that gives an excellent and accessible explanation of the origin of Pi.
http://www.amazon.com/Archimedes-Science-Living-History-Library/dp/1883937124

There's also an excellent list of math books for children here - http://www.livingmath.net/ReadersbyConcept/tabid/268/Default.aspx

I homeschool three children. My daughter 13 and my sons 11 and 9. I removed my children from public education around 2.5 years ago. When I did I realized that I should have done it sooner. Each one of my children have a strong subject and we use that to our advantage, because my weak area and theirs is math. I hate math except when it comes to budgeting, figuring out if I'm actually saving money buying precut meat or buying big slabs and cutting it at home, and if that percentage saved is going to add up to a date night with my hubby. It was an amazing thing though. My 9 year old son can look at the original price of meat and the price it is discounted to and estimate within two percent margin what we are saving. My 11 year-old can add multiple digits in his head at the grocery store and estimate the price we are checking out at within a ten dollar margin (before taxes). My 13 year-old can figure out the percentages of each discount, and then the total percentage of discounts that day and how much money that means we actually get to keep and save for a rainy day. They do it in their heads without paper and pencil (I can't even do that) and it is just from me explaining to the in the store that the less money it costs to feed them the more money we can have for other things. My children have now even started looking at coupons and determining if the coupon is worth the time and effort to clip it. My best feeling was when my children caught a deal I missed. Bags of Dole Salad were on sale for $1.49 p/bag and there were coupons in the paper that week for $1.50 off a bag. 101% savings. We walked out fo the store with all ten bags. They never enjoyed math as much as when they earned a Sunday from Dairy Queen for saving money at the grocery store. Comparing lengths and measurements was done in our kitchen while also teaching the important skill of cooking dinner for those you care about.

My five children, all young adults now, all learned early and practical math from authentic, child-driven experiences. As pre-shcool children, they learned to count by setting the table (seven plates, seven forks, etc.); they sang counting somgs, counted their toys in "cleanup games." One son, at age two, was fascinated with the numbers on the mileage gauge in the car, watching from his car seat, as the numbers rolled in order. He soon learned to count to hundred. All five kids learned to read recipes, measurements, how to divide and how to double or triple a recipe's ingredients. They read maps and figured out the mileage (long before GPS). They all played various card games and board games that utilized numbers and/or reasoning skills -- Uno, Skip-bo, Pinochle. etc. As they became involved in local sports, they learned how to keep the scorebook and figure out out averages, etc. One son learned how to make a spreadsheet to keep track of his team's batting averages. They all kept their own ledgers in their bank savings accounts.
Math, when applied to resl-life situations, can be fun and interesting. The "book math" in public school often became boring and difficult at times.
I agree with the author's observations. When a child is motivated, then add in the "readiness" component for a particular math skill, he/ she will learn the math they need at that time.
How many of us. college-graduates, use all those 12- 16 years of math? "Math" time in the early grades can be better spent on ,any other practical situatios that involved common sense, logic, reasoning, and math skills.

my personal opinion on maths in school is that they keep changing the way they teach it. for example my daughter who is 10 is being taught long division sums a totally different way from my wife and myself this means when she is stuck on a long division question it is hard for us to help her without confusing her, i just wish they would teach maths and arithmetic the old way because it works upvc windows

You're right about that. My parents were virtually unable to help me with my math assignment when I was a child and teen, and now my husband and I have trouble helping our son with his math homework. He was taught how to perform even the simplest of arithmetic exercises differently than we learned when we were young. It seems as if the methodology is changed from generation to generation in order to cause confusion in students and parents. It serves to further the feelings of math-phobia in the population.

If you focus purely on teaching a child how to do something without teaching them how to apply it then it is obviously of no use to them. On top of that, knowing how some forms of maths are applied helps people to visualise and memorise how it works. e.g. knowing that 3x3=9 would be so much easier to remember if you can visualise a square made up of 3 sides. This would also give the child a way of working out mathematics rather than simply memorising a times table.

A much more important problem is the fact that by sitting kids down with a pen and paper and trying to get them to learn something without explaining why, is just going to bore and frustrate them. Not to sound extremely cheesy but kids will certainly learn more if they can show an interest in the subject. The only way that kids will show an interest in the subject if is the subject is actually interesting.
When I was in school I used to hate maths and still do to some extent. Strangely though I can now pick up a newspaper and start playing sudoku without even considering this it is basically the same thing.

The fact is that using your brain to solve problems is fun and is something people of all ages are driven to get better at. On the other hand doing monotonous tasks is boring and will serve no purpose other than to cause people to resent education.

P.S.

Here is another point. Why do we teach how to find the dimensions of a circle yet we don't teach how to bleed radiators, put up shelves, manage finances or wire a plug? Yet again it goes back to both teaching people skills that interest them since they are in the world around them but also to teach them skills that they are almost certain to use.

For the stories of six boys from infancy to high school, learning math outside of school, as well as in school, watch the documentary Hard Problems: http://www.hardproblemsmovie.com/

Here's a comment from http://www.ted.com/talks/scott_kim_takes_apart_the_art_of_puzzles.html:
"When I started teaching my 8 home schooled children in the 1980's, I asked Scott what his advice for teaching math would be. He suggested (no surprise here!) to include lots of puzzles and games in the math curriculum. We followed his advice, buying various math manipulatives for thoughtful puzzle solving (often they were birthday presents). All our kids have excelled in math and I am ever grateful for the joy it brought to our math learning."

For anyone who'd like some free math puzzles, check out http://www.archive.org/details/amusementsinmath16713gut

This article and the 1930’s experiment by Benezet give the "what", and John Holt's classic book How Children Fail addresses the "how".

How Children Fail originated in a journal John Holt kept as he observed his own students (mostly fifth graders) and tried to puzzle out why they were capable of passing math tests (if given enough prep), but mostly incapable of using that math in real life (or even transferring it to other sorts of "school math" problems, or simply retaining it once they had finished with the test).

How Children Fail changed my life. I "dropped out" of college and went to work for Holt Associates for a time because of it.

Fast forward more than twenty years: my husband and I now have a ten-year-old and we are unschoolers. (What I mean by that is that we homeschool, but that it's our son who largely chooses what and how to learn.)

Not everyone who reads the book will be inspired to leave the world of conventional education, of course. (And I hope they don't - we need people in the schools who have read the book!)

I don’t know whether John Holt was ever aware of the experiment by Benezet (John was writing from his own experience). But I’m sure he would have been fascinated.

If you want to read How Children Fail, be sure to get the Revised Edition - it's maybe twice as long as the original (1964) edition and reflects the change in John's thinking over the following twenty years or so. Here’s a link to the Revised Edition (with a “Click to Look Inside” option):

W could repeat the experiment using a different bunch of kids: we could look at autonomously educated/unschooled kids in the UK.

In lieu of this, I'll tell you what I've seen: I've been around autonomously home educated kids now for over 16 years. Most of these children don't actively choose to study maths, (though one did want to and ended up voluntarily doing things like cube roots in his head age 7, and is heading early to Cambridge.)

The other bunch, at most, might have acquired their times tables, probably from multiplying up in Runescape or something similar. These kids go to college at 15/16 to study maths, and many thrive.

So far, most have done at least well enough to do the A levels of their choice. We have also had one who played Doom until age 16, win top prizes in maths and physics at Imperial; another, who did nothing other than read a load of fiction and manage animals in family smallholding, get into Vet school. My son has just gone to college with nothing much more than his times tables mostly under his belt, and just got 72% in his first ever maths test, which is far higher than I ever got in 10 years of private school maths.

What's more, son is also LOVING maths, which none of the others in his class, with over 10 years of school, are doing apparently.

I think maths can be acquired easily at a later stage without damage, as long as cognitive skills in other areas are developed.

What a great post! I first heard about Benezet from Alfie Kohn on Twitter. Fascinating experiment, and it is a shame more people don't know more about it.

I posted on the topic of math anxiety recently at my blog, Confessions of a Reluctant Teacher. I work as a tutor for an SAT prep company. Although I'm much stronger in verbal, I still have to cover the math portions of the SAT as well. I never took math in college, so I have only been taught as much math as they knew. Yet they would come to me with the hardest problems they encountered during practice tests, and I would almost always be able to figure them out. In my post, I write about why I think my students shut down in the face of really tough problems, and I don't--even though I did when I had to take the SAT as a high-school student.

I have subscribed to your blog and have really been enjoying it. We unschool three children. I recently wrote a post on my blog about my daughter- who by the way is a very right brained learner, and her sewing abilities and how it relates to her never being taught math. We learn what we need to know! http://pepperpaints.com/2010/02/03/but-no-one-taught-her-math/
Keep up the great work!!

I disagree with the "don't teach them math" advice. I'm still working on what I think the best solution to our dilemma (elementary teachers scared of math) is, but perhaps we should have math specialists from 1st grade on up, and have the elementary teacher learn with the kids. Maybe then in a decade or so, we'd have elementary teachers who knew how to play with math.

One of the commenters mentioned playing games. I think that's one of the best ways to learn math. Blink!, Set, Blokus are Quarto are great games that are math-related but don't focus on arithmetic skills. For that, try Yahtzee, or make up your own games with dice. Young kids also like one called Shut the Box. There are some good games at mathplayground.com, too.

Thanks for mentioning my blog. I'd like to mention the Let's Play Math! blog, and the Living Math website. Both have great resources for helping kids learn math.

I love much of what you say, Peter, but I think you're too dismissive of schools. Many families need them, and I think they could be transformed if we respected teachers and kids more.

I disagree with Sue here
"...but I think you're too dismissive of schools. Many families need them, and I think they could be transformed if we respected teachers and kids more."
I don't think we can transform schools. Teachers, especially in the younger years are respected where i live, and they still don't do a good job of 'Éducating' the children. They do a great job, however of 'schooling' them. School was never set up to truly educate, just to give enough information to people so they could do what their employer asks them to. It never has and never will consistently grow young minds who can think for themselves, sure there are some every year, who despite being 'schooled' can still think for themselves, but this is by far the abnormal. More often than not this is an outcome of their home life, not their schooling.
I do agree however that some families need school, and 'schooling' is all they require of it. That's fine, just don't get the two ideals mixed up with each other. That's why we are unhappy with the "Education System" in our societies, we think it should be doing something it was never designed to do.
Accept it as it is, if you want to have truly educated people, don't send them to school, teach them at home, in the park, in the car, at the beach, at the shops, on a bike ride, give them meaning to what they are learning, and most importantly, give them space and time to contemplate and sort it all out in their own heads. Then and only then, there will be truly educated people in our society.

"[School] never has and never will consistently grow young minds who can think for themselves, sure there are some every year, who despite being 'schooled' can still think for themselves, but this is by far the abnormal."

Look into the "factory-model education system," which details how our education model arose out of a need to train factory workers to meet certain basic prerequisites as able, obedient workers in factories. Hear that bell ringing? That's the sound of the conveyor belt starting up. Do some research and look into the history of the formation of this system that we've so readily accepted.

Yet, medicine has changed since the war, and ethical standards have been introduced.

Has our school model changed in the past 120 years? Really think about that. You're misunderstanding, though: I'm not arguing that the basis for the system is why it's wrong, I'm saying that the basis for the system tells us a lot about how this system works and the assumptions and goals that it has a foundation on.

You must remember school? It's a 30-student-per-class assembly line, and continues to be. Our system is crippling to child development. The ideal example of this is the multiple choice test. Europe has simply moved so far beyond this .. The reason why they're able to excel in maths/sciences is because they don't use a factory, multiple-choice, restricted thinking, rote memorization model like we use.

Education reform is in the air and taking root in thousands of classrooms across the country. From overhauling No Child Left Behind to closing poorly performing schools and raising student expectations, the push for change is powerful. Yet, the space where most learning takes place--the school and classroom--has changed little over the last 200 years.

You may have missed it , but following your request for documentary evidence of "fantasy play", here are some more videos of "little actors and directors":

http://vimeo.com/channels/act2cam

I hasten to add, I am able to facilitate this kind of work because I am a specially trained and very experienced teacher, who loves working with children (just like most other teachers in the school environment).

The content of the work is entirely decided by the young people, who also act and work behind the camera, under the expert guidance of two teachers.

I hope these videos serve to open your eyes to some of the emormous amount of positive, child-centric work that goes on in schools

So now I've shown you some documentary evidence of what real teaching is like, can you provide similar evidence for these following claims you make:

"Nothing [in maths teaching] has worked."
"Most of [the teachers] are math phobic"
"No matter what textbooks or worksheets or lesson plans the higher-ups devise for them, the teachers teach math by rote,"
"they just pray that no smart-alec student asks them a question such as "Why do we do it that way?" or "What good is this?"
"The students, of course, pick up on their teachers' fear, and they learn not to ask or even to think about such questions. They learn to be dumb."
"a math-schooled mind is a chloroformed mind."

Honestly Peter, for the love of God take a step back from your personal anger and look at these claims in a clear and rational manner. Are they really the well-thought-through, rational and balanced comments of a research professor of Psychology?

Peter has explained on an earlier thread about his difficulty and disappointment with his own son in state school. Unfortunately this forms his sole first-hand experience state schools, and sadly he seems reluctant to conduct any actual research into what really goes on in state schools.

I don't mind him plugging Sudbury, in which he has a personal interest (his son goes there, he has conducted research there and he is a governor there), as long as he freely admits his interests and bias.

I wouldn't mind if he ever produced some genuine contemporary research to back up his arguments (something a little more up-to-date than the 1920's anyway)

What I object to is the sustained vitriolic verbal assault on state schools, in which he resorts to his own bad experience, plus second hand accounts from other homeschoolers and research which predates the 1970s.

It is depressing to think that someone who has attained the title "research professor" has not actually done any research into state schools before attacking them in the way Peter does.

If only he stepped inside a real state school, with an open mind, he would at least have a more measured perspective.

Just a little correction, here, Steve. I have two step children who attended public (state) school, one of whom is still a student there. I have spent lots of time in such schools, including time within this year. I am sometimes invited to speak to classrooms. I have had many conversations with my step-daughter's school friends. I occasionally am invited to speak to groups of public (state) school teachers (for whom I have much respect). I am not at all unfamiliar with what happens in such schools.

While I have not felt the need to document with "studies" some of the obvious points I have made about such schools (such as the fact that they are coercive), I have in fact made use of research, including recent research, to document certain more specific points--for example, to document young people's unhappiness and boredom in school.

Also, I am not a "governor" at Sudbury Valley School. The school is governed entirely by the School Meeting, which consists of students and staff members. I am on the board of trustees, which meets once or twice a year on average to discuss broad policy issues and make suggestions. This board has absolutely no governing power. I am, unabashedly, a supporter of this type of schooling, based on my research and observations there.

Thank you for your reply, and please accept my apologies for the errors. Please would you confirm the following aspects of my comments as true:

Did you withdraw your (step)son from state school after some difficulties you had there which have left you with a lasting negative impression of state school system?

Have you conducted NO research in state schools?

As a trustee, do you have any financial connection with Sudburr?

Thank you in advance. You say you "have not felt the need to document with studies"? Well why on earth not? You say so much is obvious, yet here I am, begging for just a little solid evidence. Forget your "talks with step daughters school friends", that cuts no mustard with me. My education has taught me not to trust people just because they have a PhD. Extraordinary claims require extraordinary evidence (or at least some evidence would be nice (and you must admit, your claims that we are all coersed into schooling are somewhat extraordinary)

Evidence please. No more fudging the issue. You may not have thought contemporary research was necessary before, but I think it is your duty as a research professor to start coming up with data, and no more than 10 years old at that. Is that toom much to ask?

Steve

And don't you start repeating those claims about children's boredom in school, which I have already shown you are flawed on your previous post. Ta

Steve, I'll respond to your questions above, but this is the last time I will respond to such questions. I don't think you are interested in true discussion here.

1. I explained in an earlier post, in response to your questions, that my son left public school after 4th grade and from then on attended Sudbury Valley School. My concern for his future played a role in my initial research study of SVS--a study of the graduates. The "negative impression" I have of standard schools (not just state schools, but most private ones as well) comes from a broad range of experiences, not from the one incident that you mention. A lot of it has to do with the development of a philosophy of life--a philosophy founded in democratic values and a belief that the appropriate way to raise good democratic citizens is to present them, in their schooling, with the rights and obligations of participatory democracy. That does not happen in standard schools.

2. I have conducted no research in state schools. The focus of my research has been on self-directed learning, which doesn't happen in state schools. As you know, however, thousands of other people have and are studying learning as it occurs in state schools and almost nobody except me is studying self-directed learning. In fact, as I explained in a previous post, most studies of children of school age have been conducted in state schools and other standard school settings, and that has given us a biased understanding of children's natures.I am trying in my own small way to correct that. If you want studies of learning in state schools the education journals and book in your local university library are filled with them.

3. As a trustee I have no financial connection with SVS. This is an unpaid, elected position. I do not make any money from my connection with SVS. My interest in improving education derives from my moral concerns, my concerns with making a better world and reducing the anxiety and depression we see among so many young people today. It is in no way a financial one. I could make more money, and gain far more fame in my field, by dong other things--things that are accepted by the mainstream of current academic thought.

4. You keep asking for evidence for things that seem obvious to me and to essentially everyone else, including the many public school teachers I have talked with. The main claim that I have made that you have disagreed with repeatedly is that schools are coercive. In school children are not free. You seem to think I need evidence for that; but it is obvious to everyone else I know. If a child who is told in school that it is now time to do arithmetic responds, "No, I'm going to go outside now and climb trees," or "No, I'm going to read my Harry Potter book now," what do you think would happen? People look at the sky and see that it is blue. What is the evidence that the sky is blue? Prove it. People look at school and everyone except you sees that it is coercive. I am not going to waste time "proving" the obvious. I am going to continue to focus my research on interesting questions for which the answers are not obvious.

Thank you for taking time to respond. I genuinely appreciate it, and I am genuinely interested in discussion. My questions have a genuine purpose, and I am not intentionally trying to cause offence or annoyance. In my defence I have asked these questions before, but I don't think you you have answered them until now. See here:

http://www.psychologytoday.com/comment/reply/32931/71428

So thank you once again for answering them this time. Please bear with me while I try to answer your answers as best I can:

1.

You say

"The "negative impression" I have of standard schools... comes from a broad range of experiences, not from the one incident that you mention"

If my son was refusing to do his homework, walking out of school and was going to be put in a special needs class against my wishes, I would probably have a negative perception of the processes involved. I would see the standard education system as having failed my son, and probably rightly so. I don't think there would be any shame in you admitting that the experience was at least part of your negative view of schools.

As for your "broad range" of experiences, is it really that broad? For example, you also say that you believe

"that the appropriate way to raise good democratic citizens is to present them, in their schooling, with the rights and obligations of participatory democracy. That does not happen in standard schools."

You sound absolutely sure of this. But in my school I have a legal duty to present my kids with the rights and obligations of participatory democracy. I would bet my bottom dollar it happens in a whole bunch of US schools too. If not in your state then probably many others. It wouldn't hurt to research into this would it?

2

You say

"The focus of my research has been on self-directed learning, which doesn't happen in state schools."

PETER IT FLIPPING WELL DOES! I DO IT MYSELF! I HAVE POSTED VIDEO OF WHAT I DO AS EVIDENCE FOR YOU! AND IF YOU ACTUALLY WENT INTO A FEW MORE SCHOOLS, TALKED TO A FEW TEACHERS, INTERVIEWED A FEW KIDS, YOU WOULD KNOW, BEYOND A SHADOW OF A DOUBT, THAT SELF DIRECTED LEARNING GOES ON IN SCHOOLS! JUST DO IT, JUST GO INTO A COUPLE OF SCHOOLS, SWEET HEAVENS PLEASE!!! PLEASE!!!

Sorry for shouting. But heavens above, you can't say self-directed learning does not happen in schools if you have not done any research into schools. You say you have a "broad range of experience". Well I think you're kidding yourself. Broad range, really? How many cities you been in, how many towns you visited to come up with this mighty pronouncement? Seriously, an honest plea, visit a few schools in a few differnt counties, different states. Then come back and tell me self directed learning does not happen in schools. Show me which schools it does not go on in. I would class that as evidence

3.

You say:

"As a trustee I have no financial connection with SVS"

If your son went there, I'm guessing you paid a few thousand dollars in fees. I think that this is likely to affect your perception of the quality of the school. See here:

http://www.pnas.org/content/105/3/1050.abstract

4.

You say:

"You keep asking for evidence for things that seem obvious to me and to essentially everyone else"

I don't see evidence as a bad thing. And to be honest you haven't given me any evidence at all, have you? Maybe one survey, which you quoted as 78% of children don't like school" - Which turned out to be A LIE. Instead we found out that 28% of children like school a lot, which was a higher figure than the number of children who liked their communities. I'll say that again, the survey you quoted for me, as evidence in support of your vociferous arguments against schools, said that 28% of children like their school A LOT.

You say, "In school children are not free. You seem to think I need evidence for that"

Yes you do. Because they are free. They are free to do or not to do their homework, as your son did; they are free to leave school, as your son did. Their parents are free to send them to other schools, as your son's did. Tht, to me, is evidence of freedom.

You say, "People look at school and everyone except you sees that it is coercive."

This is not true, Peter, there are a number of people even on your site who disagree with what you are saying. I will pull up their threads for you if you insist on this particular untruth.

You say; "I am not going to waste time "proving" the obvious.

Some things which seem obvious, when held up to examination, prove false. Doesn't it seem obvious that the sun goes round the earth? Well it doesn't. A little research, a little actual scientific observation, some actual statistics, can often prove wrong something which at first seems obvious.

Listen, by all means continue with your extremely unique research into self directed learning. I genuinely find it very interesting, and that is really why I am here. I have an awful lot to offer you on self directed learning- because I do it in school. Honestly Peter, I am a trained Drama practitioner in the classroom. The children I work with have an enormous amount of freedom in their activities, with me, in school. Why don't we talk about that instead?

Only, will you please stop side tracking the issue by falsely lambasting state schools? You admit yourself that you have no evidence for your claims, nor have you actually sought any (aside from anecdotal evidence). The only actual contemporary research that you have quoted indicates that 28% of USA school children "like their schools a lot" (i admit, this is still way too low, but it is certainly not suggestive that schools are the coersive prison you are decrying them to be).

You have no need to denegrate state schools in order to promote your hunter-gatherer system- it sells itself- I would much rather talk about what you know about, and have done research on anyway, instead of stuff which you have done no research on (which also happens to be the field in which I work full time)

Yours, sincerely, and with the hope that we can at last move on to something more productive,

I did not quote a study saying that some percentage of students don't like school. Perhaps a reader did in a comment. I know nothing about that study. I did (in my Jan. 26 post) refer to a study showing that students reported themselves to be less happy, more bored, and more angry in school than in any other setting in their daily lives, but that is not the study you are referring to. The study I referred to was one that involved direct experience sampling. Now, truly this is my last response to you.
-Peter

also 94% of 34,000 Portland Parents teachers and pupils think that their schools are doing "a good job".

Sorry for thinking that you brought up these studies. They do remain quite important though. Obviously IMO some schools are doing some things right.

As for the Mihaly Csikszentmihalyi / Jeremy Hunter study you quoted in support of your position, well this following quotation comes straight from the abstract:

"Being alone rates the lowest levels of happiness, while being with friends corresponds to the highest... Youth who spend more time in school and social activities are happier than those who spend less."

Not so black and white as you make out, eh? School appears to be a good thing, it seems. I sincerely hope you don't turn your back on a sensible debate about this fact

I must agree with Steve. I would have expected more from a research professor of Psychology. Take a trip down to your math department and ask them how they feel about your article. The methods of teaching back then is drastically different from the logical hands on way math is taught today.

Whenever I've had the opportunity to broach the subject, I found that "Math for Elementary teachers" courses are some of the most dreaded and reviled courses out there: students have been described to me as being on average math-phobic and unwilling to learn. This from several sources. So there seems to be at least an anecdotal basis for that perception.

I never taught these courses, but I am quite underwhelmed by the interest for the subject displayed by most of our would be HIGH-SCHOOL math teachers, so maybe it's all for the best.

Steve, you said, "So now I've shown you some documentary evidence of what real teaching is like, can you provide similar evidence for these following claims you make:"

I know you are a teacher, but do you remember what elementary school was like when you were a kid? *I* do. *My* elementary teachers WERE math phobic. They DID teach by rote. For a teacher, you really do not know what is going on in school. Just talk to kids today and you'll get the evidence you asked for.

OK, maybe your teachers were math phobic. Does that prove Peter's point? Do you think as a consequence that all teachers are? They're not in my school, nor were they in my elementary. But that doesn't prove anything either.

Which is why we need documentary evidence.

And Peter doesn't see the need to provide his own documentary evidence. Which is why I don't think he has any. And that, IMO, is very telling

Incidentally, repetition can prove successful in some aspects of learning. Rote and routine have their place. But did your teachers teach EVERYTHING by rote? Really?

I came across the link to this article from one of my facebook friends and was particularly thrilled because just last night my blogpost included some of my thinking on this topic - http://intenselives.blogspot.com/2010/03/natural-learning-loving-it.html. My 10 year old and 5 year old are 'unschooled' and there is no way to avoid them being exposed to maths if they live a stimulating life. In particular, the computer and PS3 games that my son plays - World of Warcraft, Second Life, Uncharted, City of Heroes - have math concepts built into them in a completely natural way. I do not particularly like games that are deliberately 'educational' and my children have never liked them. Anytime I have tried to direct them to those games they lose interest very quickly perhaps because those games are often condescending in tone and less complex that a well-designed game. But give them intelligent games to play and almost inevitably they learn lots of things that we are trying to get them to cover in the school syllabus, and they learn them in a much more natural and effortless way.
But I think that any real effort to remove formalized mathematics from the curriculum and you will have a major outcry. In my exposure to lots of unschooled families (and of course this is anecdotal) I have found that the dads often have a really hard time agreeing to leave out direct maths teaching. I have no idea why but I have seen it repeatedly and often more strident if the child is a boy. On several online lists it becomes a bitter subject in custody hearings. So the people to tackle first if any change is to take place would be the parents.
Anyhow, thanks again for this delightful post. I'll add it to my blog.
GG

I remember all through school how upset everyone would get when presented with a "word problem" that was more than just some digits lined up in nice algorithmic columns. Students just weren't equipped to figure them out. First you did 400 "real" math problems and then you were supposed to figure out how those problems related to two trains, one leaving Chicago at 4pm and one leaving New York at 2am.

I was a top math student all through school but I was able to compartmentalize my desire to know how or why stuff worked and just do what they told me. My absolute favorite math area was geometry because it was so clear how it worked but most other students hated geometry the most probably exactly because it wasn't a bunch of numbers in nice rows.

Now I have 3 kids attending a democratic free school with no imposed curriculum. And like Gillian above my kids have spent a lot of time playing online games. Real games, not those stupid educational ones. My 11 year old son who has been out of "real school" for 3 years plays MapleStory and has figured out complex mathematical structures just to play the game. If I want to buy this helmet for this amount how many hours do I have to play making this amount per hour in order to buy the helmet? If I sell this item in the market and the fee to sell is a certain percent how much will I have left after the fee? If I have this percentage of experience and I make a certain percentage per hour of experience how many hours will it take to level up and if I kill this creature that gives these points what percentage will each creature give? If I have this level of health what health potion will be most economical to buy? Plus in the game you work with three different currencies and have to be able to translate back and forth among them regularly. Put these problems isolated from the game context to a bunch of 5th graders in real school and ask them to show their work and see what you get.

Another game we're playing online is Roblox, a variation on the Lego concept. It's geometry start to finish. In fact, that's a lot of how I actually learned to understand numbers: blocks and Lego.

My oldest daughter manages a 130 horse virtual stable online and has to figure out how to make and spend enough money to get all the stuff she wants without killing her horses. Plus she has to put together html code to get stuff to work. What is code but problems laid out in a row. This instruction plus that instruction equals that result. If the middle isn't right then the answer is wrong and the code doesn't work.

Another important concept that is inherent in online game play is time zones. If your game server is based in Sweden and all your ponies die at the 0 hour in Sweden what time is that in Eastern Standard Time? Try putting that one on a math test.

On a more real-world level all three kids, 9, 11 and 14 get a set amount of money to spend a week on whatever they choose. They can spend it on lunch or eat free school lunch. They can spend it all on virtual money for their games or buy candy or save some for later, bigger purchases. They have to go buy the stuff themselves and deal with change and receipts. Believe me, figuring out how much you can spend on chips and how much on game cards is a major motivation to figure out how many quarters are in a dollar.

Plus I talk out loud when I'm figuring out how many cups of water it takes to make 3 packages of rice if one package is 2/3 of a cup. In fact I talk out loud about any math problem I'm figuring out. Which box of sandwich bags is a better deal? How many hours until something happens. Math is the air we move in every day and my three kids who are essentially unschooled are all absorbing how numbers work just by living.

The sad thing is my oldest who spent the longest in real school is still convinced from those days that she's math stupid. She had a great intuitive understanding of math concepts but she didn't get the rote learning thing that is inherent in how we teach math in school. She just couldn't do the 400 problems all lined up in tidy meaningless rows. The trick is how to tie the math she knows now just from existing in a world of numbers back into her actually knowing that she does know math.

My conclusions after observing my very small and unscientific sample population is that math instruction teaches you very little but how to pass math tests but that the real world is full of math that kids actually want to learn. How many Snickers CAN you buy for $15 anyway?

From personal experience I've found that public schools have sort of a knack for stifling a child’s curiosity. When I came out of home school and in to public school in the fifth grade, all of my classmates had little or no interest in learning and most of them really hated every aspect of school other then the social interaction with other classmates they were able to enjoy. The main reason for this, I believe, is that most of the classes in that school were either extremely dull or required too much memorization without any sort of interactivities. The simple fact is, if a class is boring a child won't want to be in it and won't learn anything.

I'm not saying we have to entertain children to educate them, but we have to find a way to make subjects appeal to children in order to spark their curiosity and we have to base the teaching of more abstract subjects, such as algebra and calculus, back to real world situations that students can relate to. If not, students will quickly lose interest and those with little self-discipline or parental encouragement will simply flunk out.

I'm in high school now, and regretfully must of my classmates struggle with basic subjects; I can't help but think it somehow has to do with the failure of our educational system.

What is wrong with entertaining children to educate them? As mentioned earlier, games are one of the best ways to help children learn math, and I'm a big believer in the idea that play is the best way that children learn--playing is all *about* learning.

I don't think that entertainment for the sake of entertainment is necessarily a good thing, especially in an educational setting, but as a tool to engage children in the subject matter, it can make a big difference.

Thought I would contribute our experience of not teaching any formal maths to autonomously educated son - until he chose to do it at age 12.

We really did do virtually nothing until he was 12. He then shot through key stage 3 maths in about 3 weeks of doing just a little bit here and there. He found it almost ridiculously easy, doing things like memorising his tables (with some interest in the various patterns that he spotted) in less than an hour. Trigonometry easy peasy, equations no problem.

OK so he had spent a considerable amount of his younger years playing the markets on Runescape and solving other mathematical problems in various (fun) games...but that had been it.

SOO much less sweat.

We had found the same thing with learning to read. He started late, aged 8, but was somehow overnight an adult level reader. Again so much less sweat.

Here in Denmark, Hasseris Gymnasium, Aalborg, I have taught maths. Well, I haven't. I chose not to. Pupils come into the gymnasium at 15-17 years old. We use a superb text, solid maths covering many areas of value for higher studies. They work through the text over the year. No formal instruction is given to the class. No previous information on pupils is referred to. It is a new start. They also mark there own answers and they know that it is a moral issue. Do the work, read the text, mark the answers afterwards. Be honest, do not cheat as you cheat yourself. The we have 20 minute tests every week or two. There final grade covers these test marks. They turn up!!

Pupils cooperate, consult others, move into teams, study at home and ask me questions. I spend most of my time at the board helping with specific queries. Now pupils use the board to help each other.

Parents are baffled. Some are simply stunned. Success is evident and a previous trial in another school revealed great progress (7-15 years old). A team of Japanese professors on a visit were impressed beyong belief. They could not accept my role in the class. So what have I learnt?

Quite simply, by being in the class I allow certain freedoms. Why do some pupils struggle? At present five out of 25 pupils are having difficulty. The primary explanation is lack of study discipline. Often I see MP3 players, sometimes a computer is present to help the maths (of course it is not), weaker pupils do not use stronger pupils to help them and they sit together with others who struggle. Some use a calculator on a phone or a computer. In short their issues seem to have been a complete lack of study techniques and approaches. These pupils are not using the right tools, other suitable pupils, silent time to study and often are late of absent (doing little at home).

In a couple of cases there is a poor mathematical background, they have been led to believe they are weak or not guided in study skills. This is a confidence issue. I play it down.

The pupils in the same class last year were formally taught, given little to do at home and covered far less material. We know it fails and I can see it. I know that I have been given total freedom to do this by my Rector. No other employer, except my last one, would tolerate this. I think it is trivial but I am aware this is very controversial.

The Rector is elated, the pupils happy and the parents often amazed at the commitment of their children.

I am happy to write more if there is interest. I can be contacted by interested parties. Visits are welcome.

I am a teacher in a state school and would love to know more, particularly about your findings about the kids who do "little at home", and how you find it affects their progress.

I have a strong feeling that people need to know about the incredible diversity of education that goes on in schools. In short, it isn't the uniform prison that Peter is makes out and people deserve to hear the honest truth from profesionals who actually work successfully with large numbers of kids on a day to day basis.

We have student self-marked work in the UK too. We call it "assessment for learning". I'd be interested to know if there was something similar in the US

Basically if we place emphasis on pupils to maintain a pace it can be successful but the secret is getting the balance right. If you let them progress and mark their won work we see that some can regulate their efforts by keeping an eye on the others. Of 32 students 15-18 years old we lost five due to lack of middle school academic rigour and now three to five more are behind. They admit to not working at home and clearly look for reasons- social activity, procrastination, poor study skills, lack of suitability. They know they have a commitment problem.

Some have raced ahead, test well and engage. This is around 5-10 pupils. In general I see guilt playing a role, parental expectations, poverty and general home life (no space, peace and quiet). Good students have conscience and ethical positions. The messages I give to them are positive and supportive. They have to know the reality. Often they will go home and the parents are in full agreement with my views. It helps to have academic parents, parents who are teachers or wealth.

We should talk further. I basically run a "Grand National" style with the fences as tests and the in-between ground as their responsibility.

Catch me on stewartbonephd@hotmail.com

I have worked in 7 schools, two in the UK. Now I am in the IB Diploma system in a danish gymnasium (well paid, low hours, respectable job, excellent resources). It is very relaxed but you can perform on your own agenda. Naturally we are being impacted by the bad economic conditions and students are streaming our way. The middle school system in denmark is in trouble and we are picking up the problems. The me generation and spoilt kids are legion and they simply have no idea how brutal it can be to fail or underachieve. It worries me as I can't see this generation handling what is coming up and it is not pretty. Being poor in denmark isn't really poor but it can be depressive for many and they struggle to move forward.